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Properties

Label 2-165-11.9-c1-0-2
Degree $2$
Conductor $165$
Sign $0.758 - 0.651i$
Analytic cond. $1.31753$
Root an. cond. $1.14783$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.288 + 0.887i)2-s + (0.809 − 0.587i)3-s + (0.912 + 0.663i)4-s + (−0.309 − 0.951i)5-s + (0.288 + 0.887i)6-s + (1.65 + 1.19i)7-s + (−2.36 + 1.71i)8-s + (0.309 − 0.951i)9-s + 0.933·10-s + (1.85 − 2.75i)11-s + 1.12·12-s + (−0.447 + 1.37i)13-s + (−1.54 + 1.11i)14-s + (−0.809 − 0.587i)15-s + (−0.145 − 0.446i)16-s + (0.267 + 0.824i)17-s + ⋯
L(s)  = 1  + (−0.203 + 0.627i)2-s + (0.467 − 0.339i)3-s + (0.456 + 0.331i)4-s + (−0.138 − 0.425i)5-s + (0.117 + 0.362i)6-s + (0.624 + 0.453i)7-s + (−0.835 + 0.606i)8-s + (0.103 − 0.317i)9-s + 0.295·10-s + (0.558 − 0.829i)11-s + 0.325·12-s + (−0.124 + 0.382i)13-s + (−0.411 + 0.299i)14-s + (−0.208 − 0.151i)15-s + (−0.0362 − 0.111i)16-s + (0.0649 + 0.200i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.758 - 0.651i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.758 - 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24457 + 0.461140i\)
\(L(\frac12)\) \(\approx\) \(1.24457 + 0.461140i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-1.85 + 2.75i)T \)
good2 \( 1 + (0.288 - 0.887i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (-1.65 - 1.19i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.447 - 1.37i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.267 - 0.824i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.53 - 1.83i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 4.70T + 23T^{2} \)
29 \( 1 + (1.64 + 1.19i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.27 + 10.0i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.36 + 2.44i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.651 + 0.473i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 2.34T + 43T^{2} \)
47 \( 1 + (8.39 - 6.10i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.22 + 6.86i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.73 - 4.89i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.70 + 8.33i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 3.15T + 67T^{2} \)
71 \( 1 + (-3.97 - 12.2i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-11.9 - 8.65i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (5.25 - 16.1i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.89 - 5.84i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 3.77T + 89T^{2} \)
97 \( 1 + (0.598 - 1.84i)T + (-78.4 - 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.90898920405893805055613010511, −11.88425567062747670413656163173, −11.29743380823892812215403310217, −9.556420043997190542258997601193, −8.367901614145528037257846966876, −8.071639582324156821352774414299, −6.67509823126948937184511971449, −5.69148880131465756875569464499, −3.87716268859030780995415636749, −2.13950264487069106687506346923, 1.85142206062892420411106955866, 3.33076794014860897793317903996, 4.76096274738198933000221498110, 6.49423286257637952562001108899, 7.51410245032711732399237013484, 8.831169279467373855067390118925, 10.02871145418494330703519492366, 10.55860749252030184960118319139, 11.59039158261367233701593827728, 12.43191734804448553235136088682

Graph of the $Z$-function along the critical line